jeremywatco,
Is the Non-Mitel system integration via SIP Trunks or SIP Extensions?
My assumption is that they are SIP Extensions, which is the easiest way, but I can't assume that since you are not able to access the trunks with Dial 8/9.
If they are SIP Trunks then you have the problem of making Trunk to Trunk calls through the system.
Thanks,
TE